Time Series Analysis

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Transcript of Time Series Analysis

Time Series AnalysisConclusionIn summary, time series analysis represents an important field of statistics and is used in various fields of application.

"The best qualification of a prophet is to have a good memory."

Marquis of Halifax Advantages and Disadvantagesof Time Series Analysis in General Classical Component Model of Time Series AnalysisSpecific Models of Time Series AnalysisTypical Examples of a Time SeriesClassical Component Modelof Time Series AnalysisDaily closing prices of market-listed derivativesMonthly measured unemployment ratesAnnual production rates of a steel plantQuarterly revenue of a companyTime Series Analysis in GeneralClassical Component Model of Time Series AnalysisA time series is a sequence of observed values of a certain characteristic, which are chronologically in succession and mostly periodical among the same carrier.Definition of Time SeriesAdvantagesValuable to identify seasonal variationsOften practiced method, probably more then theoretical methodsDisadvantagesThere are no completely true componentsValidity of hypotheses are only partially verifiableComponents have to be functions of timeMoving averageExponential smoothingTrend extrapolationBox-Jenkins

Moving AverageAim: calculate an average for the trend of historic data and smooth out a time seriesDifferentiation into: simple, exponential and weighted moving averageArithmetic mean of defined number of successive values within a specific time frameCategorization into moving average of even and uneven order

Application of the Moving AverageMoving average of uneven order:Moving average of even order:Application of the Moving AverageDatabase for Example in LectureAdvantages and Disadvantages of the Moving AverageAdvantagesSimple, practicable, and effective in useBasic mathematical tools are sufficientEasy in terms of comprehensibilityDisadvantagesDifficult to decide about number of considered valuesSimple moving average states an equal weighting for all valuesIt is suggested that the time series has an regular cycleExponential SmoothingMathematical-statistical method for short-term to maximum medium-term horizonsDifferentiation into: single, double and triple exponential smoothingRecent observed values are more meaningful for the prediction of the future than earlier observed

Values are exponentially increasing from past to present

Application of Exponential SmoothingApplication of Exponential SmoothingAdvantages and Disadvantages of Exponential SmoothingTrend ExtrapolationSimplest way to define forecastsExtrapolation relates to a single time seriesHistoric data with regularities is continued into future

--> Trends in data must be valid for future as wellBased on the regularities in the historic data trends are extrapolated into the future Application of Trend ExtrapolationPossibility to use the free-hand method by sense of proportionMathematical way is more appropriateDifferent mathematical approaches--> linear, parabolic, exponential and logistic trend, as well as gompertz curve

Application of Trend ExtrapolationBox-Jenkins methodology Box-Jenkins methodologyOrigin: George Box and Gwilym Jenkins in 1970 Main condition for the application: stationarity (mean variance and autocorrelation function that are essentially constant through time)Principle of parsimony: The more parameters to estimate, the more errors.Aim: finding a good model that describes how observations in a time series are related to each other

Comparison to classical modelConsideration of stochastic processes instead of deterministic processes in order to model a time series. Classical model: Cyclical movements are modeled as stationary processes around the deterministic trend. Box-Jenkins: random shocks are considered and can have a permanent effect on subsequent time series data.

Basic Box-Jenkins ProcessesAutoregressive Process AR (p)Basic Box-Jenkins ProcessesAutoregressive Moving Average Process ARMA (p, q)The Box-Jenkins model building processSingle Exponential Smoothing:AdvantagesSimple, robust, easy to usePossible to utilize very short time seriesNot an art to update if new data is availableDisadvantagesOverly simplistic and inflexibleNot optimal for capturing any linear dependence in dataHard to forecast in the long run as values are highly influenced by recent happenings in the history of the time seriesDatabase for Example in LectureApplication of Exponential SmoothingMoving Average Process MA (q)Treatment of non-stationary processes: ARIMA (p, d, q)Application of the trend extrapolation based on the linear trend states a mathematical functionAdditiveMultiplicativeHybrid formAdvantages and Disadvantages of Trend ExtrapolationAdvantagesSimple to useTime as only element to change in functionCan be used to define short-, medium- and long-term forecastsDisadvantagesFluctuations in business cycle are not consideredIn a practical view it is no longer sufficient to extrapolate trendsRegularities of the past are suggested to be valid in future as wellSmooth ComponentAdvantagesWide spectrum of software programs enables simulation of various types of models with correspondent resultsConsideration of random shocks which have a permanent effect on subsequent time series dataReduction of errors due to the principle of parsimonyDisadvantagesDifficult interpretation of the modelNeed of time series with a minimum of 50 observations Modeling process requires high investment of time and resources to build a satisfactory model Database for Example in LectureApplication of Trend ExtrapolationAdvantages and Disadvantages of the Box-Jenkins